16.3

16.3 The Fundamental

Theorem for Line Integrals

The Fundamental Theorem for Line Integrals

We know that Part 2 of the Fundamental Theorem of Calculus can be written as

where F′ is continuous on [a, b].

We also called Equation 1 the Net Change Theorem:

The integral of a rate of change is the net change.

The Fundamental Theorem for Line Integrals

If we think of the gradient vector ∇f of a function f of two or three variables as a sort of derivative of f, then the following theorem can be regarded as a version of the Fundamental Theorem for line integrals.

Example 1

Find the work done by the gravitational field

in moving a particle with mass m from the point (3, 4, 12) to the point (2, 2, 0) along a piecewise-smooth curve C.

Solution:

We know that F is a conservative vector field and, in fact, F = ∇f, where

Example 1 – Solution

Therefore, by Theorem 2, the work done is W =

∫

∫

= f(2, 2, 0) – f(3, 4, 12)

cont’d

Independence of Path

Independence of Path

Suppose C₁ and C₂ are two piecewise-smooth curves

(which are called paths) that have the same initial point A and terminal point B.

We know that, in general,

∫

∫

∫

∫

whenever ∇f is continuous. In other words, the line integral of a conservative vector field depends only on the initial

point and terminal point of a curve.

Independence of Path

In general, if F is a continuous vector field with domain D, we say that the line integral

∫

path if

∫

∫

points.

With this terminology we can say that line integrals of conservative vector fields are independent of path.

A curve is called closed if its terminal point coincides with its initial point, that is,

r(b) = r(a). (See Figure 2.) Figure 2

A closed curve

Independence of Path

If

∫

followed by the path C₂ from B to A. (See Figure 3.)

Figure 3

Independence of Path

Then

∫

∫

∫

∫

∫

since C₁ and –C₂ have the same initial and terminal points.

Conversely, if it is true that

∫

Take any two paths C₁ and C₂ from A to B in D and define C to be the curve consisting of C₁ followed by –C₂.

Independence of Path

Then

0 =

∫

∫

∫

∫

∫

and so

∫

∫

Thus we have proved the following theorem.

Since we know that the line integral of any conservative vector field F is independent of path, it follows that

Independence of Path

The physical interpretation is that the work done by a conservative force field as it moves an object around a closed path is 0.

The following theorem says that the only vector fields that are independent of path are conservative. It is stated and proved for plane curves, but there is a similar version for space curves.

Independence of Path

We assume that D is open, which means that for every

point P in D there is a disk with center P that lies entirely in D. (So D doesn’t contain any of its boundary points.)

In addition, we assume that D is connected: this means that any two points in D can be joined by a path that lies in D.

The question remains: how is it possible to determine

whether or not a vector field F is conservative? Suppose it is known that F = P i + Q j is conservative, where P and Q have continuous first-order partial derivatives. Then there is

Independence of Path

Therefore, by Clairaut’s Theorem,

The converse of Theorem 5 is true only for a special type of region.

Independence of Path

To explain this, we first need the concept of a simple curve, which is a curve that doesn’t intersect itself

anywhere between its endpoints. [See Figure 6; r (a) = r (b) for a simple closed curve, but r(t₁) ≠ r(t₂) when

a < t₁ < t₂ < b.]

Independence of Path

In Theorem 4 we needed an open connected region. For the next theorem we need a stronger condition.

A simply-connected region in the plane is a connected region D such that every simple closed curve in D

encloses only points that are in D.

Notice from Figure 7 that, intuitively speaking, a

simply-connected region contains no hole and can’t

consist of two separate pieces.

Figure 7

Independence of Path

In terms of simply-connected regions, we can now state a partial converse to Theorem 5 that gives a convenient

method for verifying that a vector field on is conservative.

Example 2

Determine whether or not the vector field

F (x, y) = (x – y) i + (x – 2) j is conservative.

Solution:

Let P (x, y) = x – y and Q(x, y) = x – 2. Then

Since ∂P/∂y ≠ ∂Q/∂x, F is not conservative by Theorem 5.

Conservation of Energy

Conservation of Energy

Let’s apply the ideas of this chapter to a continuous force field F that moves an object along a path C given by r(t), a ≤ t ≤ b, where r(a) = A is the initial point and r(b) = B is the terminal point of C.

According to Newton’s Second Law of Motion, the force F(r(t)) at a point on C is related to the acceleration

a(t) = r″ (t) by the equation

F(r(t)) = m r″(t)

So the work done by the force on the object is

Conservation of Energy

Therefore

where v = r′ is the velocity.

(Fundamental Theorem of Calculus)

(By formula d/dt [u(t)  v(t)]

= u′(t)  v(t) + u(t)  v′(t))

Conservation of Energy

The quantity that is, half the mass times the square of the speed, is called the kinetic energy of the object. Therefore we can rewrite Equation 15 as

W = K(B) – K(A)

which says that the work done by the force field along C is equal to the change in kinetic energy at the endpoints of C.

Now let’s further assume that F is a conservative force field;

that is, we can write F = ∇f.

Conservation of Energy

In physics, the potential energy of an object at the point (x, y, z) is defined as P(x, y, z) = –f(x, y, z), so we have F = –∇P.

Then by Theorem 2 we have

W =

∫

∫

= P(A) – P(B)

Conservation of Energy

Comparing this equation with Equation 16, we see that P(A) + K(A) = P(B) + K(B)

which says that if an object moves from one point A to

another point B under the influence of a conservative force field, then the sum of its potential energy and its kinetic

energy remains constant.

This is called the Law of Conservation of Energy and it is the reason the vector field is called conservative.